# Is there a mathematical description for the states of consistency between two sets?

Is there a mathematical description for the various states of consistency two sets can be in relative to each other? I can draw out the various set states I can think of, but I would like to have some way of proving that no other states existed.

*Edit - Here is a list of the states I am referring to:

1. Set A and Set B are equal.
2. Set A and Set B are not equal, but A is a subset of B
3. Set A and Set B are not equal, but B is a subset of A
4. Set A and Set B are not equal, but they have an intersection and they have an equal number of elements that are not in common with each other e.g. Set A has 2 elements not in Set B, and Set B has 2 elements not in Set A
5. Set A and Set B are not equal, but they have an intersection and they have an unequal number of elements that are not in common with each other e.g. Set A has 3 elements not in Set B, and Set B has 2 elements not in Set A
6. Set A and Set B are not equal and have no intersection

What I am looking for is a way to express this in set notation and to discover if I am missing some states.

• Can you describe the details of how you create your diagrams? Right now, I don't really understand the notation in your images. – Michael Burr Dec 10 '15 at 19:31
• Apologies - I have no experience in this. In the first diagram I mean there are 'E' elements in common between the two sets, but Set A has '1+N' additional elements. In the second diagram, it is the other way around. In the third diagram they have 'E' elements in common, and both have the same number of elements that are not in common. – user297670 Dec 10 '15 at 19:35
• What is a "state of consistency"? Are there states of inconsistency? Are you asking how many ways there are for two sets to intersect? – John Douma Dec 10 '15 at 19:42
• I was using consistency/inconsistency in layman's terms which might have confused matters - I regarded two sets as consistent if they are equal. All other states are inconsistent. Asking "how many ways there are for two sets to intersect" is probably a better way of putting it. – user297670 Dec 10 '15 at 19:49
• You can do what you're asking, but it is important for you to include all the details of what you mean by the various states. You are asking for more than how many ways are there for two sets to intersect (or at least "ways" is unclear). It seems that you're asking, for two sets $A$ and $B$, whether $|A|=|B|$, $|A\cap B|=0$, $|A\setminus B|>|B\setminus A|$, etc. – Michael Burr Dec 10 '15 at 19:55